Re: SUO: An SUO 'Bookshelf' of Results
Mike,
I'd be happy to facilitate transfer of summaries to the web site. When
step 4 in your process is reached, I'd suggest that the author send me a
message with a subject line like "Summary for web site posting".
Adam
At 10:59 AM 3/15/2001 -0800, Michael Uschold wrote:
>Everyone,
>
>I have not been following the details of the triadicity discussion, but I was
>delighted to see Pat Hayes' detailed explication of it. I'm guessing that
>it is
>to a significant extent, a condensed and synthesized view of this thread so
>far. When this issue dies down, and there is reasonable agreement on things,
>including what to disagree on, then it would be possible to record this as an
>'official' outcome of the SUO group.
>
>I propose that a kind of SUO bookshelf be set up on our web pages. It would
>contain detailed technical summaries similar to what Pat has produced. This
>would be an
>
> EFFECTIVE VEHICLE FOR DISSEMINATION OF THE
> SIGNIFICANT PROGRESS WE HAVE MADE ON MANY FRONTS,
>
>which to date is virtually inaccessible, buried in the discussion archives.
>One can think of this as real-time knowledge mining of our archives.
>This would affect many folk:
>
>1. The world at large can see our progress to date.
>2. Newcomers can much more easily get up to speed.
>3. Regulars can skip the details and be assured of a good summary, in time.
>4. All participants can be more efficient in the time they spend reading SUO,
> being able to keep up to speed on things on their own time, without fear
> of missing key information.
>
>Personally, I have on many occasions been on the verge of getting off this
>list
>because it is overwhelming. Yet, there are MANY things that are of great
>interest and relevance to me, which I lack the time to follow and
>contribute in
>detail to. A kind of SUO bookshelf, I hope would make everyone make more
>efficient use of their SUO-time.
>
>I propose the following process/format for how this might happen.
>
>1. Discussion threads take place in the usual fashion.
>
>2. When things settle down a bit, and points of agreement are reached, these
>should be summarized and fed back to the group. This would also include points
>where people agree to disagree.
>
>3. When the main participants in the discussion reach agreement on the
>summary, then it could be submitted to the overall group for placement
>on the 'bookshelf'. The assumption is that things DO go on the bookshelf.
>I don't think we want a formal voting process for this. The idea of going to
>the group at large, is to see if there are any further suggestions or
>improvements to the document.
>
>4. We announce the existence of each new entry to the bookshelf to a SUO
>interest group which is only intersted in the outcomes and timely
>announements,
>rather than the day to day discussions. We may already have such list, I do
>not know whether the creation of SUO sub-lists some months ago, actually
>worked.
>
>I would hope or expect that in many cases, these summaries could be the basis
>for quality technical papers that are publishable.
>
>Can we submit a journal arcitle with SUO as author? :-)
>
>Mike Uschold
>------------
>
> >From owner-standard-upper-ontology@ieee.org Wed Mar 14 17:01:21 2001
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>To: standard-upper-ontology@ieee.org
>From: pat hayes <phayes@ai.uwf.edu>
>Subject: SUO: irreducible triadicity
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>
>I know I said I wouldnt say any more on this topic here, but several
>people have asked me for the review I mentioned, so here is a brief
>summary of the pith and essence, shorn of references to Burch's
>formalism and terminology.
>
>On irreducible triadicity.
>
>There are three distinct domains to consider. The first is a purely
>mathematical notion of a connected graph; the second is formal
>relational languages; and the third, somewhat murkier, is concerned
>with the metaphysical nature of relations. Peirce proved a simple
>result in graph theory to which he and others attribute metaphysical
>significance; but it does so only if one chooses a particular
>connection between the first and third of our domains via the second.
>Other connections are possible, under which the result, while of
>course still true, seems to have much less significance.
>
>The result in graph theory is, roughly, that any connected graph can
>be built using nodes of degree 1,2 and 3, but that one cannot do it
>with nodes only of degree 1 and 2. (Degree of a node is the number
>of arcs connected to it.) More precisely, it is that every connected
>graph can be 'expanded' to a graph built using such nodes, where one
>expands a graph by replacing some of its nodes by connected subgraphs
>whose external arcs are isomorphic to the arcs to that node in the
>first graph. In effect, the expanded graph has a subgraph which can
>be collapsed (by identifying all its nodes and any connections
>between them) into a single node of the original graph. In modern
>terms one might say that the expanded graph is an implementation of
>the first graph in which some nodes are implemented by datastructures
>which are themselves connected graphs: imagine using LISP to encode a
>graph, for example.
>
>It is easy to see how to construct such a datastructure for any node
>of degree n >3; one simply takes n-2 degree-3 nodes and links them
>together in a chain: (view the following in a fixed-width font)
>
> | | | | |
> ---1------2------3------4---....---n-2---
>
>which has n outgoing arcs to which the nodes linked to the original
>node can be attached. Also it is pretty obvious that you couldn't do
>it with nodes of degree 2, since the process of linking them together
>would itself use up all but the two spare 'ends', so you can only get
>another degree-2 graph.
>
>That is the sum total of the notion of 'irreducible triadicity'. It
>is an interesting result in graph theory, but why is it considered to
>have any deep importance in semantics and semiotics? To see why
>Peirce thought it did, one has to appreciate how Peirce would have
>interpreted a semantic network.
>
>A semantic network is a way of interpreting labelled graphs as sets
>of assertions: the nodes of the graph correspond to things, and the
>arcs of the graph correspond to relations. Only binary relations can
>be directly accomodated in this way, so to encode the fact that a
>relation R of greater arity (eg a trinary relation) holds, one
>introduces a new node to be an 'R-fact', and links it to the
>arguments of the trinary relation by arcs (which can be labelled with
>relations like 'first', 'second', etc., if no intuitively sensible
>relation names suggest themselves.) This corresponds to the logic
>translation I gave in an earlier message, and it has been widely used
>in KR work for many years. (To give a full translation of FOL into
>graphs requires other devices, including some way to represent
>quantifier scope. John Sowa's CG's are an excellent example; I will
>ignore this from now on as it is orthogonal to the 'triadicity'
>issue.)
>
>With this encoding of relational language into graphs, the
>irreducible triadicity result is not important, since the degree of a
>node plays no special role in the interpretation (it corresponds to
>the number of times a name is used in a logical expression). Peirce
>however had a rather different way of interpreting such a graph. In
>Peirce's graphical notation, the nodes of the graph, not the arcs,
>are thought of as indicating relations, so that the degree of the
>node is the number of arguments the relation has. With this
>interpretation the degree of a node is clearly of much greater
>importance, and the graph-theoretic result takes on a new
>significance. It is therefore interesting to investigate this other
>'Peircian' way of interpreting of a graph.
>
>One might object immediately that if the nodes are relations, what
>part of the graph indicates the things the relations hold between, ie
>the things related by the relations? This question gets to the heart
>of the Peirce/Whitehead notion of the world being in some sense made
>of process, rather than things; 'things' are thought of here as a
>kind of convenient illusion (one which arises, in fact, from noticing
>relations.) The relations are seen as the the metaphysical ground on
>which the notion of individual is itself built. I do not want here to
>get involved in this metaphysical discussion, but will just remark
>that it is completely divorced from what might be called the
>intellectual mainstream of the last century, in which mathematics and
>formal semantics have been based on set theory, which in turn is
>rooted in the idea of collecting together things, a notion which
>depends on the idea of one individual thing being distinct from
>another. I mention all this only to partly motivate what would
>otherwise seem to be a very odd answer to the question about what in
>the the graph denotes the individuals, which is Peirce's answer:
>nothing. There are no individuals being related; there are only
>relations. The basic 'connection' between relations expressed by the
>arcs in the graph is now rather mysterious, but it can be thought of
>as a kind of existential connection: it says that the two relations
>share a kind of factual bonding at this point; they are mutually
>instantiated. In modern terms one would write this as an explicit
>existential claim using a quantifier:
>(exists x)(R1(...x...) & R2(.....x...))
>where the dots indicate that the other arguments are filled in with
>other, different, variables.
>The quantifier is not really needed, and one can just use 'anonymous'
>names of free variables:
>R1(...x...) & R2(.....x...)
> This is what an arc in the graph translates into in Peirce's view of
>graphs as assertions. Notice that this is an exact dual of the
>semantic network view of the graph: here, the arcs are the 'things'
>and the nodes are the relations.
>
>If one thinks of relations as atoms, then this bonding is rather like
>chemical valency, and the connected graphs which result are analogous
>to molecules; and logic becomes a kind of relational chemistry. This
>metaphor is superficially attractive, and in particular it has the
>merit, if one feels that these 'things' are best kept out of sight,
>of disposing of the connections as real things in themselves.
>
>This view of what a graph means, however, has some limitations.
>Notice that in the translation into logic sketched above, it is
>possible to use any variable (or individual name) at most twice,
>corresponding to the two ends of the arc in the graph. This produces
>a curiously attenuated logic, in which for example it is not possible
>to say that something has three properties:
>P(x) & Q(x) & R(x)
>has no Peircian graph corresponding to it (it would need an arc with
>three ends). To overcome this, Peirce introduces a special class of
>relations, called identity relations. Then the graph which would say
>that this P-Q-R-ish thing existed would have the following
>translation:
>P(x) & Q(y) & R(z) & I3(x, y, z) (notice each variable is only in 2 places)
>where I3 is the special relation of three-way identity, which Peirce
>called 'teridentity'. The graph looks like this:
>
> P
> |
> |
> R----I3 ----Q
>
>There can be such special relations I-n for all finite n, but in fact
>we only need I3, since the others can be 'implemented' using I3 in
>the way outlined at the beginning, by chaining enough (n-2) I3's
>together. Using a modern translation, for example, I5 can be defined
>as
>
>I5(x y z u v) <==> I3(x y A) & I3(A z B) & I3(B u v)
>
>where the A and B links are 'private' to this little subgraph. Peirce
>attributed great significance to the identity relations, as well he
>might, since once they are put in place they clearly play the role of
>individuals. To assert a relation of identity is to say that
>something exists, and linking the identity to other relations says
>that they hold of thing that exists.
>
>The above reduction-to-I3 trick doesnt work if you try to use just
>I2, ie good old equals. You might think that it would be easy, since
>it is easy to write it in modern logic:
>I3(x y z) <==> (x=y) & (y=z)
>but if you now try to use this rewrite, the 'y' has been used up
>since it occurs twice in the definition already - no name can be used
>more than twice - so you can't say that anything else equals y; so
>this is really just I2(x,z). The two-ended nature of graph arcs has
>got you cornered. Triadicity really is irreducible in Peircian graph
>language.
>
>Notice that the only triadic relation we really need is I3 itself,
>since we can string together a suitable implementation for any n-ary
>node using copies of I3, and even string an extra link to a node
>which holds the label of the original node as well, and use binary or
>unary relations for everything else; so any claim that some
>*particular* relation (other than I3) is itself irreducibly triadic
>must be based on some other criterion.
>
>The interest of this irreducibility, however, is relative to how
>seriously one views the metaphysical consequences of the Peircian
>interpretation of graphs. I suggest that it isn't of much interest
>(other than historical), for several reasons. First, there's an
>obviously better interpretation available (semantic networks.)
>Second, the triadicity result applies to graphs, but it doesnt apply
>to a simple generalization which is just as mathematically
>respectable, if harder to draw, and which provides a more natural
>structure to interpret in the Peircian fashion. Hypergraphs are
>graphs where an arc (called a hyperarc) can link more than two nodes.
>These have a very direct Peircian-style interpretation which doesnt
>require the rather artificial 'identity relation' nodes; but
>triadicity is reducible in hypergraphs. Third, if one asks what the
>'identity relation' really is saying, it is clear that it amounts to
>what would be expressed in modern terms as an existential assertion,
>ie it says that some THING exists; and if we can refer to that thing,
>it is obvious that any n-fold identity can be expressed as a
>conjunction of binary equality statements. The irreducibility of
>triadicity arises from a curiously obtuse combination of insisting
>that existence can be expressed by using an identity relation, and
>refusing to allow any way of referring to the thing that exists.
>
>Pat Hayes
>
>---------------------------------------------------------------------
>IHMC (850)434 8903 home
>40 South Alcaniz St. (850)202 4416 office
>Pensacola, FL 32501 (850)202 4440 fax
>phayes@ai.uwf.edu
>http://www.coginst.uwf.edu/~phayes
>
>--============_-1227507994==_ma============
>Content-Type: text/enriched; charset="us-ascii"
>
>I know I said I wouldnt say any more on this topic here, but several
>people have asked me for the review I mentioned, so here is a brief
>summary of the pith and essence, shorn of references to Burch's
>formalism and terminology.
>
>
>On irreducible triadicity.
>
>
>There are three distinct domains to consider. The first is a purely
>mathematical notion of a connected graph; the second is formal
>relational languages; and the third, somewhat murkier, is concerned
>with the metaphysical nature of relations. Peirce proved a simple
>result in graph theory to which he and others attribute metaphysical
>significance; but it does so only if one chooses a particular
>connection between the first and third of our domains via the second.
>Other connections are possible, under which the result, while of course
>still true, seems to have much less significance.
>
>
>The result in graph theory is, roughly, that any connected graph can be
>built using nodes of degree 1,2 and 3, but that one cannot do it with
>nodes only of degree 1 and 2. (Degree of a node is the number of arcs
>connected to it.) More precisely, it is that every connected graph can
>be 'expanded' to a graph built using such nodes, where one expands a
>graph by replacing some of its nodes by connected subgraphs whose
>external arcs are isomorphic to the arcs to that node in the first
>graph. In effect, the expanded graph has a subgraph which can be
>collapsed (by identifying all its nodes and any connections between
>them) into a single node of the original graph. In modern terms one
>might say that the expanded graph is an implementation of the first
>graph in which some nodes are implemented by datastructures which are
>themselves connected graphs: imagine using LISP to encode a graph, for
>example.
>
>
>It is easy to see how to construct such a datastructure for any node of
>degree n >3; one simply takes n-2 degree-3 nodes and links them
>together in a chain: (view the following in a fixed-width font)
>
>
><fixed> | | | | |
>
> ---1------2------3------4---....---n-2---
>
>
></fixed>which has n outgoing arcs to which the nodes linked to the
>original node can be attached. Also it is pretty obvious that you
>couldn't do it with nodes of degree 2, since the process of linking
>them together would itself use up all but the two spare 'ends', so you
>can only get another degree-2 graph.
>
>
>That is the sum total of the notion of 'irreducible triadicity'. It is
>an interesting result in graph theory, but why is it considered to have
>any deep importance in semantics and semiotics? To see why Peirce
>thought it did, one has to appreciate how Peirce would have interpreted
>a semantic network.
>
>
>A semantic network is a way of interpreting labelled graphs as sets of
>assertions: the nodes of the graph correspond to things, and the arcs
>of the graph correspond to relations. Only binary relations can be
>directly accomodated in this way, so to encode the fact that a relation
>R of greater arity (eg a trinary relation) holds, one introduces a new
>node to be an 'R-fact', and links it to the arguments of the trinary
>relation by arcs (which can be labelled with relations like 'first',
>'second', etc., if no intuitively sensible relation names suggest
>themselves.) This corresponds to the logic translation I gave in an
>earlier message, and it has been widely used in KR work for many years.
> (To give a full translation of FOL into graphs requires other devices,
>including some way to represent quantifier scope. John Sowa's CG's are
>an excellent example; I will ignore this from now on as it is
>orthogonal to the 'triadicity' issue.)
>
>
>With this encoding of relational language into graphs, the irreducible
>triadicity result is not important, since the degree of a node plays no
>special role in the interpretation (it corresponds to the number of
>times a name is used in a logical expression). Peirce however had a
>rather different way of interpreting such a graph. In Peirce's
>graphical notation, the nodes of the graph, not the arcs, are thought
>of as indicating relations, so that the degree of the node is the
>number of arguments the relation has. With this interpretation the
>degree of a node is clearly of much greater importance, and the
>graph-theoretic result takes on a new significance. It is therefore
>interesting to investigate this other 'Peircian' way of interpreting of
>a graph.
>
>
>One might object immediately that if the nodes are relations, what part
>of the graph indicates the things the relations hold between, ie the
>things related by the relations? This question gets to the heart of the
>Peirce/Whitehead notion of the world being in some sense made of
>process, rather than things; 'things' are thought of here as a kind of
>convenient illusion (one which arises, in fact, from noticing
>relations.) The relations are seen as the the metaphysical ground on
>which the notion of individual is itself built. I do not want here to
>get involved in this metaphysical discussion, but will just remark that
>it is completely divorced from what might be called the intellectual
>mainstream of the last century, in which mathematics and formal
>semantics have been based on set theory, which in turn is rooted in the
>idea of collecting together things, a notion which depends on the idea
>of one individual thing being distinct from another. I mention all this
>only to partly motivate what would otherwise seem to be a very odd
>answer to the question about what in the the graph denotes the
>individuals, which is Peirce's answer: nothing. There are no
>individuals being related; there are only relations. The basic
>'connection' between relations expressed by the arcs in the graph is
>now rather mysterious, but it can be thought of as a kind of
>existential connection: it says that the two relations share a kind of
>factual bonding at this point; they are mutually instantiated. In
>modern terms one would write this as an explicit existential claim
>using a quantifier:
>
>(exists x)(R1(...x...) & R2(.....x...))
>
>where the dots indicate that the other arguments are filled in with
>other, different, variables.
>
>The quantifier is not really needed, and one can just use 'anonymous'
>names of free variables:
>
>R1(...x...) & R2(.....x...)
>
> This is what an arc in the graph translates into in Peirce's view of
>graphs as assertions. Notice that this is an exact dual of the semantic
>network view of the graph: here, the arcs are the 'things' and the
>nodes are the relations.
>
>
>If one thinks of relations as atoms, then this bonding is rather like
>chemical valency, and the connected graphs which result are analogous
>to molecules; and logic becomes a kind of relational chemistry. This
>metaphor is superficially attractive, and in particular it has the
>merit, if one feels that these 'things' are best kept out of sight, of
>disposing of the connections as real things in themselves.
>
>
>This view of what a graph means, however, has some limitations. Notice
>that in the translation into logic sketched above, it is possible to
>use any variable (or individual name) at most twice, corresponding to
>the two ends of the arc in the graph. This produces a curiously
>attenuated logic, in which for example it is not possible to say that
>something has three properties:
>
>P(x) & Q(x) & R(x)
>
>has no Peircian graph corresponding to it (it would need an arc with
>three ends). To overcome this, Peirce introduces a special class of
>relations, called identity relations. Then the graph which would say
>that this P-Q-R-ish thing existed would have the following
>translation:
>
>P(x) & Q(y) & R(z) & I3(x, y, z) (notice each variable is only in 2
>places)
>
>where I3 is the special relation of three-way identity, which Peirce
>called 'teridentity'. The graph looks like this:
>
>
><fixed> P
>
> |
>
> |
>
> R----I3 ----Q
>
>
></fixed>There can be such special relations I-n for all finite n, but
>in fact we only need I3, since the others can be 'implemented' using I3
>in the way outlined at the beginning, by chaining enough (n-2) I3's
>together. Using a modern translation, for example, I5 can be defined as
>
>
>I5(x y z u v) <<==> I3(x y A) & I3(A z B) & I3(B u v)
>
>
>where the A and B links are 'private' to this little subgraph. Peirce
>attributed great significance to the identity relations, as well he
>might, since once they are put in place they clearly play the role of
>individuals. To assert a relation of identity is to say that something
>exists, and linking the identity to other relations says that they hold
>of thing that exists.
>
>
>The above reduction-to-I3 trick doesnt work if you try to use just I2,
>ie good old equals. You might think that it would be easy, since it is
>easy to write it in modern logic:
>
>I3(x y z) <<==> (x=y) & (y=z)
>
>but if you now try to use this rewrite, the 'y' has been used up since
>it occurs twice in the definition already - no name can be used more
>than twice - so you can't say that anything else equals y; so this is
>really just I2(x,z). The two-ended nature of graph arcs has got you
>cornered. Triadicity really is irreducible in Peircian graph language.
>
>
>Notice that the only triadic relation we really need is I3 itself,
>since we can string together a suitable implementation for any n-ary
>node using copies of I3, and even string an extra link to a node which
>holds the label of the original node as well, and use binary or unary
>relations for everything else; so any claim that some *particular*
>relation (other than I3) is itself irreducibly triadic must be based on
>some other criterion.
>
>
>The interest of this irreducibility, however, is relative to how
>seriously one views the metaphysical consequences of the Peircian
>interpretation of graphs. I suggest that it isn't of much interest
>(other than historical), for several reasons. First, there's an
>obviously better interpretation available (semantic networks.) Second,
>the triadicity result applies to graphs, but it doesnt apply to a
>simple generalization which is just as mathematically respectable, if
>harder to draw, and which provides a more natural structure to
>interpret in the Peircian fashion. Hypergraphs are graphs where an arc
>(called a hyperarc) can link more than two nodes. These have a very
>direct Peircian-style interpretation which doesnt require the rather
>artificial 'identity relation' nodes; but triadicity is reducible in
>hypergraphs. Third, if one asks what the 'identity relation' really is
>saying, it is clear that it amounts to what would be expressed in
>modern terms as an existential assertion, ie it says that some THING
>exists; and if we can refer to that thing, it is obvious that any
>n-fold identity can be expressed as a conjunction of binary equality
>statements. The irreducibility of triadicity arises from a curiously
>obtuse combination of insisting that existence can be expressed by
>using an identity relation, and refusing to allow any way of referring
>to the thing that exists.
>
>
>Pat Hayes
>
>---------------------------------------------------------------------
>
>IHMC (850)434 8903 home
>
>40 South Alcaniz St. (850)202 4416 office
>
>Pensacola, FL 32501 (850)202 4440 fax
>
>phayes@ai.uwf.edu
>http://www.coginst.uwf.edu/~phayes
>
>--============_-1227507994==_ma============--
-----------------
Adam Pease
Teknowledge
(650) 424-0500 x571